Stick-Slip Interface Motion as a Singular Limit of the Viscosity-Capillarity Model

Abstract

This work is a follow-up on a study by Vainchtein and Rosakis of interface dynamics and hysteresis in materials undergoing solid-solid phase transitions. The author investigates the dynamics of a bar with a nonconvex double-well elastic energy density. The model includes both viscosity and strain-gradient capillarity terms. Viscous stress provides eiergy dissipation. The capillarity term models interfacial energy. The bar is subject to time-dependent displacement boundary conditions. Numerical simulations predict hysteretic behavior in the overall load-elongation diagram. The hysteresis is primarily due to metastability and persists even at very slow loading when viscous dissipation is small. At a given loading, a large capillarity coefficient a results in a smooth interface motion and small hysteresis loop. As a becomes smaller, the loop grows and acquires serrations, while the interface motion alternates between slow and fast regimes. The results suggest that the stick-slip interface motion and serrated hysteresis loop predicted by Vainchtein and Rosakis in the absence of interfacial energy are a singular limit of the viscosity-capillarity model as the capillarity coefficient tends to zero. The irregular interface motion and serrated load-elongation curves qualitatively agree with some experimental observations in shape-memory alloys.